This survey covers numerous results of the last 20 years concerning the multiplicative groups of integral group rings as well as classical K- theory. The bibliography includes 189 titles. The following contents reflects the variety of topics considered in the survey. Note that V(\({\mathbb{Z}}G)\) denotes the normalized multiplictive group of the integral group ring \({\mathbb{Z}}G\) of a group G. {\S}1. Elements of finite order of the group V(\({\mathbb{Z}}G)\). {\S}2. Multiplicative group of a commutative integral group ring. {\S}3. Triviality of elements of finite order and triviality of the multiplicative group of a group ring. {\S}4. Periodic normal subgroups of the multiplicative group of a group ring. {\S}5. Group-theoretic properties of the multiplicative group with trivial elements of finite order. {\S}6. Free subgroups of the multiplicative group of a group ring. {\S}7. The unitary subgroup of the multiplicative group of a group ring. {\S}8. Congruence-subgroups of the multiplicative group of a group ring. {\S}9. Conjugacy of finite subgroups in the multiplicative group of a group ring. {\S}10. Matrix representations and generators of the multiplicative group of a group ring. {\S}11. Projective modules and elements of classical K-theory. {\S}12. Projective modules over integral group rings of finite groups. {\S}13. Projective modules over group rings of almost polycyclic groups.